The floor of the Pacific Ocean is divided into several plates. The largest one, the Pacific Plate, is moving northwest relative to the plate that holds North America, and too hot spots are coming up through the mantle from below the plates (they generate islands like Hawaii).
On the earth's surface, nothing can move along a straight line. Instead, the plates rotate around a point on the surface. The pacific plate rotates around a point south of Australia. Around Hawaii, the plate moves at about 7 cm/year, or about as fast as fingernails grow.
The evidence for this motion is pretty convincing:
Earthquakes: earthquakes occur on the boundaries of the plates as they rub past each other. The motion observed during these earthquakes agrees well with the expected motion.
Absolute measurements: Thanks to the Global Positioning System navigation system, we can now measure the location of any point on the earth within a few cm. The observed motions agree with the expected plate motions.
Plate boundaries: where two plates meet, there can be three types of boundaries: transform faults where the plates slide past each other (like the San Andreas fault); spreading centers where two plates are moving away from each other (like the Juan de Fuca spreading center off the Pacific NW and the mid-Atlantic Ridge; and subduction zones where one plate dives into the earth under another plate (Aleutian Islands, Japan). All the observed boundaries should agree with the expected motions, and they do.
Magnetism: new rock is added to the plates at spreading centers as they split apart. As this lava cools, it becomes magnetized, much like a magnetic tape in your tape recorder. When the magnetic field of the earth changes, this change is recorded in the rocks. We can read this record by sensing the changes in the magnetic field from ships, and we see stripes of different magnetism parallel to the spreading centers, as expected if the plates are splitting apart there.
Sediments: in the deep ocean, there is a very slow but continuous rain of mud that slowly settles to the ocean's bottom and forms layers of sediment. Since the ocean floor is the youngest near the spreading centers, we would expect very little sediment there, and the layers should get thicker the farther you get from a spreading center. This is observed, and the age of the oldest (deepest) fossils in the sediment obtained by drilling in the ocean floor gives us an idea of the age of the ocean floor at that spot. Studies show that the ocean floor gets older as you get farther away from spreading centers and that the oldest ocean floor today is only about 200 million years old. That may sound very old, but the oldest fossils are over 3 billion years old.
There is more evidence, and you can read about it in a good geology textbook.